Download Statistical Model Selection Methods Applied to Biological Networks

Statistical Model Selection Methods Applied
to Biological Networks
Michael P.H. Stumpf1, , Piers J. Ingram1 , Ian Nouvel1 , and Carsten Wiuf2
1
2
Centre for Bioinformatics, Department of Biological Sciences, Wolfson Building,
Imperial College London, London SW7 2AZ, UK
Bioinformatics Research Center, University of Aarhus, 8000 Aarhus C, Denmark
[email protected]
Abstract. Many biological networks have been labelled scale-free as
their degree distribution can be approximately described by a powerlaw
distribution. While the degree distribution does not summarize all aspects of a network it has often been suggested that its functional form
contains important clues as to underlying evolutionary processes that
have shaped the network. Generally determining the appropriate functional form for the degree distribution has been fitted in an
ad-hoc fashion.
Here we apply formal statistical model selection methods to determine
which functional form best describes degree distributions of protein interaction and metabolic networks. We interpret the degree distribution
as belonging to a class of probability models and determine which of
these models provides the best description for the empirical data using
maximum likelihood inference, composite likelihood methods, the Akaike
information criterion and goodness-of-fit tests. The whole data is used
in order to determine the parameter that best explains the data under a
given model (e.g. scale-free or random graph). As we will show, present
protein interaction and metabolic network data from different organisms
suggests that simple scale-free models do not provide an adequate description of real network data.
1
Introduction
Network structures which connect interacting particles such as proteins have
long been recognised to be linked to the underlying dynamic or evolutionary
processes[13, 3]. In particular the technological advances seen in molecular biology and genetics increasingly provide us with vast amounts of data about
genomic, proteomic and metabolomic network structures [15, 22, 19]. Understanding the way in which the different constituents of such networks, — genes
and their protein products in the case of genome regulatory networks, enzymes
and metabolites in the case of metabolic networks (MN), and proteins in the
case of protein interaction networks (PIN) — interact can yield important insights into basic biological mechanisms [16, 24, 1]. For example the extent of
Corresponding author.
C. Priami et al. (Eds.): Trans. on Comput. Syst . Biol. III, LNBI 3737, pp. 65–77, 2005.
c Springer-Verlag Berlin Heidelberg 2005
66
M.P.H. Stumpf et al.
phenotypic plasticity allowed for by a network, or levels of similarity between
PINs in different organisms presumably depend on topological (in a loose sense
of the word) properties of networks.
Our analysis here focuses on the degree distribution of a network, i.e. the
probability of a node to have k connections to other nodes in the network. While
it is well known that this does not offer an exhaustive description of network
data, it has nevertheless remained an important characteristic/summary statistic
of network data. Here we use Pr(k) to denote a theoretical model for the degree
distribution, or Pr(k; θ) if the model depends on an (unknown) parameter θ (potentially vector-valued), and P̂r(k) to denote the empirical degree distribution.
Many studies of biological network data have suggested that the underlying
networks show scale-free behaviour [8] and that their degree distributions follow
a power-law, i.e.
(1)
Pr(k; γ) = k −γ /ζ(γ)
where ζ(x) is Riemann’s zeta-functions which is defined for x > 1 and diverges
as x → 1 ↓; for finite networks, however, it is not necessary that the value of γ
is restricted to values greater than 1.
These powerlaws are in marked contrast to the degree distribution of the
Erdös-Rényi random graphs [7] which is Poisson, Pr(k; λ) = e−λ λk /k!. The study
of random graphs is a rich field of research and many important properties can be
evaluated analytically. Such Poisson random networks (PRN) are characterized
by most nodes having comparable degree; the vast majority of nodes will have
a connectivity close to the average connectivity.
The term ”scale-free” means that the ratio Pr(αk)/Pr(k) depends on α alone
but not on the connectivity k. The attraction of scale-free models stem from the
fact that some simple and intuitive statistical models of network evolution cause
powerlaw degree distribution. Scale-free networks are not, however, the only type
of network that produces fat-tailed degree distributions.
Here we will be concerned with developing a statistically sound approach
for inferring the functional form for the degree distribution of a real network.
We will show that relatively basic statistical concepts, like maximum likelihood
estimation and model selection can be straightforwardly applied to PIN and MN
data. In particular we will demonstrate how we can determine which probability
models best describe the degree distribution of a network. We then apply this
approach in the analysis of real PIN data from five model organisms and MN
data. In each case we can show that the explanatory power of a standard scalefree network is vastly inferior compared to models that take the finite size of the
system into account.
2
Statistical Tools for the Analysis of Network Data
Here we are only concerned with methods aimed at studying the degree distribution of a network. In particular we want to quantify the extent to which a given
functional form can describe the degree distribution of a real network. Given a
probability model (e.g. power-law distribution or Poisson distribution) we want
Statistical Model Selection Methods Applied to Biological Networks
67
to determine the parameters which describe the degree distribution best; after
that we want to be able to distinguish which model from a set of trial model
provides the best description. Here we briefly introduce the basic statistical concepts employed later. These can be found in much greater detail in most modern
statistics texts such as [12]. Tools for the analysis of other aspects of network
data, e.g. cluster coefficients, path length or spectral properties of the adjacency
matrix will also need to be developed in order to understand topological and
functional properties of networks.
There is a well established statistical literature that allows us to assess to
what extent data (e.g. the degree distribution of a network) is described by a
specific probability model (e.g. Poisson, exponential or powerlaw distributions).
Thus far, determining the best model appears to have been done largely by eye
[13] and it is interesting to apply a more rigorous approach, although in some
published cases maximum likelihood estimates were used to determine the value
of γ for the scale-free distribution.
2.1
Maximum Likelihood Inference
Since we only specify the marginal probability distribution, i.e. the degree distribution, we take a composite likelihood approach to inference, and treat the
degrees of nodes as independent observations. This is only correct in the limit
of an infinite sized network and finite sized sample (n << N , where N denotes
network size and n the sample size). Composite likelihood methods are becoming
increasingly popular in cases for which the full likelihood is difficult to specify
and/or the full likelihood is intractable to calculate numerically. In our case
Table 1. Network models and their degree distributions, Pr(k; θ). Wherever it appears,
C denotes the normalizing constant such that k Pr(k; θ) = 1.
Degree distribution Pr(k; θ)
k
exp(−λ) λk!
for all k ≥ 0
C exp(−k/k̄)
for all k ≥ 0
kγ−1 e−k
for
all k ≥ 0
Γ (γ)
0
for k = 0
Network type
Poisson
Exponential
Gamma
Scale-free
Truncated scale free network
Scale-free network
with exponential cut-off
Lognormal
Stretched exponential
k−γ /ζ(γ)
0
−γ M
k / i=L k−γ
0
for k > 0
for k < L and k > M
for L ≤ k ≤ M
for k < k0 and k > kcut
(k + k0 )−γ exp(−k/kcut ) for k0 ≤ k ≤ kcut
− ln((k−θ)/m)2 /(2σ2 )
Ce
(k−θ)σ
√
2π
for all k ≥ 0
0
for k < 0
C exp(−αk/k̄)k−γ
for k > 0
Model
M1
M2
M3
M4
M4a
M4b
M5
M6
68
M.P.H. Stumpf et al.
the full likelihood is difficult to specify. Reference [11] provides an overview of
composite likelihood methods.
For a given functional form or model Pr(k; θ) of the degree distribution we
can use maximum likelihood estimation applied to the composite likelihood in
order to estimate the parameter which best characterizes the distribution of
the data. The composite likelihood of the model given the observed data K =
{k1 , k2 , . . . , kn } is defined by
L(θ) =
n
Pr(ki ; θ),
(2)
i=1
and taking the logarithm yields the log-likelihood
lk(M ) = lk(θ) =
n
log(Pr(ki ; θ)).
(3)
i=1
The maximum likelihood estimate (MLE), θ̂, of θ is the value of θ for which
Eqns. (2) and (3) become maximal. For this value the observed data is more
probable to occur than for any other parameters.
Here the maximum likelihood framework is applied to the whole of the data.
This means that in fitting a curve —such as a powerlaw k −γ̂ /ζ(γ̂), where γ̂
denotes the MLE of the exponent γ— data for all k is considered. If a powerlawdependence where to exist only over a limited range of connectivities then the
global MLE curve may differ from such a localized power-law (or equivalently
any other distribution).
2.2
Model Selection and Akaike Weights
We are interested in determining which model describes the data best. For nonnested models (as are considered here, e.g. scale-free versus Poisson) we cannot
use the standard likelihood ratio test but have to employ a different information
criterion to distinguish between models: here we use the Akaike-information
criterion (AIC) to choose between different models [2, 10]. The AIC for a model
Pr(k; θ) is defined by
AIC = 2(−lk(θ̂) + d),
(4)
where θ̂ is the maximum liklihood estimate of θ and d is the number of parameters required to define the model, i.e. the dimension of θ. Note that the
model is penalized by d. The model with the minimum AIC is chosen as the best
model and the AIC therefore formally biases against overly complicated models.
A more complicated model is only accepted as better if it contains more information about the data than a simpler model. (It is possible to formally derive
the AIC from Kohn-Sham information theory.) Other information criteria exist,
e.g. the Bayesian information criterion (BIC) offers a further method for penalizing more complex models (i.e. those with more parameters) unless they have
significantly higher explanatory power (see [10] for details about the AIC and
Statistical Model Selection Methods Applied to Biological Networks
69
model selection in statistical inference). In order to compare different models we
define the relative differences
∆AIC
= AICj − min(AIC),
j
j
(5)
where j refers to the jth model, j = 1, 2, . . . , J, and minj is minimum over all
j.This in turn allows us to calculate the relative likelihoods (adjusted for the
dimension of θ) of the different models, given by
/2).
exp(−∆AIC
j
(6)
Normalizing these relative likelihoods yields the so-called Akaike weights wj ,
exp(−∆AIC
/2)
j
wj = J
.
AIC
/2)
j=1 exp(−∆j
(7)
The Akaike weight wj can be interpreted as the probability that model j
(out of the J alternative models) is the best model given the observed data and
the range of models to choose from. The relative support for one model over
another is thus given by the ratio of their respective Akaike weights. If a new
model is added to the J existing models then the analysis has to be repeated.
The Akaike weight formalism is very flexible and has been applied in a range of
context including the assessment of confidence in phylogenetic inference [20]. In
the next section we will apply this formalism to PIN data from five species and
estimate the level of support for each of the models in table 1.
2.3
Goodness-of-Fit
In addition to the AIC or similar information criteria we can also assess a model’s
performance at describing the degree distribution using a range of other statistical measures. The Kolmogorov-Smirnoff (KS)[12] and Anderson-Darling (AD)
[4, 5] goodness-of-fit statistics allow us to quantify the extent to which a theoretical or estimated model of the degree distribution describes the observed data.
The former is a common and easily implemented statistic, but the latter puts
more weight on the tails of distributions and also allows for a secular dependence
of the variance of the observed data on the argument (here the connectivity k).
They KS statistic is defined as
D = max |P̂ (k) − P (k)|,
(8)
where P̂ (k) and P (k) are the empirical and theoretical cumulative distribution
functions, respectively, for a node’s degree i.e. P (k) = ki=1 Pr(i) and P̂ (k) =
k
k
i=1 P̂r(i). If P (k) depends on θ, P (k) is substituted by P (k; θ̂) =
i=1 Pr(i; θ̂),
the estimated cumulative distribution. This statistic is most sensitive to differences between the theoretical (or estimated) and observed distributions around
the median of the data, i.e. the point where P (k) ≈ 0.5. Given that we will
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M.P.H. Stumpf et al.
also be considering a number of fat-tailed distributions this is somewhat unsatisfactory and we will therefore also use the AD statistic (Anderson and Darling
discussed a number of statistics [4, 5]) which is defined as
|P̂ (k) − P (k)|
D∗ = max P (k)(1 − P (k))
(9)
(again P (k) might be substituted for P (k; θ̂)).
We can use these statistics for two purposes: first, we can use them to compare
different trial distributions as the ”best” distribution should have the smallest
value of D and D∗ , respectively. Second, we can use these statistics to determine
if the empirical degree distribution is consistent with a given theoretical (or
estimated) distribution.
To evaluate the fit of a model, p-values can be calculated for the observed
values of D and D∗ using a parametric boot-strap procedure using the estimated
degree distribution: for a network with N nodes we repeatedly sample N values
at random from the maximum likelihood model, Pr(k, θ̂) and calculate D∗ and
D∗ , respectively, for each replicate. From L bootstrap-replicates we obtain the
Null distribution of D and D∗ under the estimated degree distribution. For
sufficiently large L we can thus determine approximate p-values which allow us
to test if the empirical degree distribution is commensurate with the estimated
degree distribution.
3
Statistical Analysis of Biological Networks
Here we apply the analysis of the preceeding sections to the study of PINs and
metabolic networks. It is easy to find a straight-line fit to some degree interval
for all of the datasets considered here. For a powerlaw to be meaningful it has
to extend over at least two or three decades and with a maximum degree of
kmax 300 this will be unachievable for the present data sets. We therefore use
all the data and fit the model which yields the best overall a description of the
degree distribution.
3.1
Analysis of PIN Data
In table 2 we show the maximum composite likelihoods for the degree distributions calculated from PIN data collected in five model organisms [23] (the protein interaction data was taken from the DIP data-base; http://dip.doe-mbi.
ucla.edu). We find that the standard scale-free model (or its finite size versions)
never provides the best fit to the data; in three networks (C.elegans, S.cerevisiae
and E.coli) the lognormal distribution (M5) explains the data best. In the remaining two organisms the stretched exponential model provides the best fit to
the data. The bold likelihoods correspond to the highest Akaike weights. Apart
from the case of H.pylori (where max(wj ) = w5 ≈ 0.95 for M5 and w6 ≈ 0.05)
the value of the maximum Akaike weight is always > 0.9999. For C elegans,
Statistical Model Selection Methods Applied to Biological Networks
71
however, the scale-free model and its finite size versions are better than the
lognormal model, M5.
Table 2. Log-likelihoods for the degree distributions from table 1 in five model organisms. M1, M2, M3 and M4 have one free parameter, M4a, M4b and M5 have two free
parameters, while M6 has three free parameters. The differences in the log-likelihoods
are, however, so pronounced that the different numbers of parameters do not noticeably
influence the AIC.
Organism
D.melanogaster
C.elegans
S.cerevisiae
H. pylori
E.coli
M1
-38273
-9017
-24978
-2552
-834
M2
-20224
-5975
-14042
-1776
-884
M3
-29965
-6071
-20342
-2052
-698
M4
-18517
-4267
-13454
-1595
-799
M4a
-18257
-4258
-13281
-1559
-789
M4b
-18126
-4252
-13176
-1546
-779
M5
-17835
-4328
-12713
-1527
-659
M6
-17820
-4248
-12759
-1529
-701
33
1
2
3
5
8
16
n(k)
56
111 210
535
1248
S.cerevisiae
1
2
3
4
6
8 11
16 23 33 47 67 95 141 220
k
Fig. 1. Yeast protein interaction data (o) and best-fit probability distributions: Poisson
(—), Exponential (—), Gamma (—), Power-law (—), Lognormal (—), Stretched
exponential (—). The parameters of the distributions shown in this figure are the
maximum likelihood estimates based on the real observed data.
For the yeast PIN the best fit curves (obtained from the MLEs of the parameters of models M1-M6) are shown in figure 1, together with the real data.
Visually, log-normal (green) and stretched exponential (blue) appear to describe
the date almost equally well. Closer inspection, guided by the Akaike weights,
however, shows that the fit of the lognormal to the data is in fact markedly better than the fit of the stretched exponential. But the failure of quickly decaying
72
M.P.H. Stumpf et al.
distributions such as the Poisson distribution, characteristic for classical random
graphs [7] to capture the behaviour of the PIN degree distribution is obvious.
Interestingly, common heuristic finite size corrections to the standard scalefree model improve the fit to the data (measured by the AIC). But compared to
the lognormal and stretched exponential models they still fall short in describing
the PIN data in the five organisms.Figure 2 shows only the three curves with
Table 3. The KS and AD statistics D and D∗ for the scale-free, lognormal and
stretched exponential model. The ordering of these models is in agreement with the
AIC, but KS and AD statistics capture different aspects of the degree distribution. We
see that the likelihood treatment, which takes in all the data, agrees better with the
KS statistic. In the tails (especially for large connectivities k) the maximum likelihood
fit sometimes —especially in the case of E.coli— can provide a very bad description of
the observed data.
Species
M4
D D∗
D.melanogaster 0.13 0.26
C.elegans
0.03 0.09
S.cerevisiae 0.17 0.33
H. pylori
0.13 0.26
E.coli
0.28 0.56
M5
D D∗
0.01 0.06
0.10 0.20
0.01 0.04
0.01 0.05
0.04 56.09
M6
D D∗
0.02 0.06
0.02 0.08
0.03 5.99
0.02 0.12
0.12 6072
the highest values of ωj , which apart from E.coli are the log-normal, stretched
exponential and power-law distributions; for E.coli, however, the Gamma distribution replaces the power-law distribution. These figures show that, apart from
C.elegans the shape of the whole degree distribution is not power-law like, or
scale-free like, in a strict sense. Again we find that log-normal and stretched exponential distributions are hard to distinguish based on visual assessment alone.
Figures 1 and 2, together with the results of table 2, reinforce the well known
point that it is hard to choose the best fitting function based on visual inspection.
It is perhaps worth noting, that the PIN data is more complete for S.cerevisiae
and D.melanogaster than for the other organisms.
The standard scale-free model is superior to the log-normal only for C.elegans.
The order of models (measured by decreasing Akaike weights) is M6, M5, M4,
M2, M3, M1 for D.melanogaster, M6, M4, M5, M2, M3, M1 for C.elegans, M5,
M6, M4, M2, M3, M1 for S.cerevisiae and H.pylori, and M5, M3, M6, M4, M2,
M1 for E.coli. Thus in the light of present data the PIN degree distribution of
E.coli lends more support to a Gamma distribution than to a scale-free (or even
stretched scale-free) model. There is of course, no mechanistic reason why the
gamma distribution should be biologically plausible but this point demonstrates
that present PIN data is more complicated than predicted by simple models.
Therefore statistical model selection is needed to determine the extent to which
simple models really provide insights into the intricate architecture of PINs. For
Statistical Model Selection Methods Applied to Biological Networks
D.melanogaster
3
1
1
2
4
7 12 22 40 71 137
1
2 3
5
8
14 25 45 80 153
k
k
E.coli
H.pylori
16
1
1
2
3
6
4 7
13
n(k)
30
53
73
140
173
1
n(k)
7 18 53 176
n(k)
20 56 176
3
7
n(k)
750
1541
C.elegans
73
1
2
3
5 7
k
11
18 28 44
1
2
3
5 7
11
18 28 44
k
Fig. 2. Degree distributions of the protein interaction networks (o) of C.elegans,
D.melanogaster, E.coli and H.pylori. The power-law (—), log-normal (—) and
stretched exponential (—) models are shown for all figures; for E.coli the gamma
distribution (—), which performs better (measured by the Akaike weights) than either
scale-free and the stretched exponential distributions.
completeness we note that model selection based on BIC results in the same
ordering of models as the AIC shown here.
In table 3 we give the values of D and D∗ for the empirical degree distributions. The estimated cumulative distribution P (k; θ̂) was obtained from the
maximum likelihood fits of the respective models. The results in table 3 show
that the maximum likelihood framework (or the respective models) sometimes
cannot adequately describe the tails of the distribution —at low and high values
of the connectivity— in some cases, where D∗ >> D. The order of the different
models suggested by D for the three models generally agrees with the ordering
obtained from the AIC.
74
3.2
M.P.H. Stumpf et al.
Analysis of MN Data
Metabolic networks aim to describe the biochemical machinery underlying cellular processes. Here we have used data from the KEGG database (www.genome.
jp/kegg) with additional information from the BRENDA database (www.
brenda.uni-koeln.de). The nodes are the enzymes and an edge is added between two enzymes if one uses the products of the other as educts.
Table 4. Log-likelihoods for the degree distributions from table 1 applied to metabolic
networks. Human and yeast data were extracted from the global database.
Data
KEGG
H.sapiens
S.cerevisiae
M1
-8276
-1619
-2335
M2
-5247
-1390
-1485
M3
-7676
-1565
-2185
M4
-5946
-1525
-1621
M5
-5054
-1312
-1436
M6
-4178
-1308
-1427
13
1
2
4
6
9
n(k)
22 32
57
88
168
Metabolic network
1
2
3
4
5
7
9
12 16 21 28 37 49 65
k
Fig. 3. Degree distributions of the metabolic network data (o) and best-fit probability
distributions: Poisson (—), Exponential (—), Gamma (—), Power-law (—), Lognormal (—), Stretched exponential (—). The parameters of the distributions shown
in this figure are the maximum likelihood estimates based on the real observed data.
Ignorig low degrees (k = 1, 2) when fitting the scale-free model increases the exponent (i.e. it falls off more steeply) but compared to the other models no increased
performance is observed.
From figure 3.2 and table 4 it is apparent that the maximum likelihood
scale-free model obtained from the whole network data does not provide an
Statistical Model Selection Methods Applied to Biological Networks
75
adequate description of the MN data. This should, however, not be too surprising
as the network is only relatively small with maximum degree kmax = 83. The
degree distribution appears to decay in an essentially exponential fashion but
the stretched exponential has the required extra flexibility to describe the whole
of the data better than the other models. Measured by goodness of fit statistics
D and D∗ , however, the log-normal model (D = 0.02 and D∗ = 0.06) performs
rather better than the stretched exponential (D = 0.07 and D = 0.22); the
scale-free model again performs very badly (D = 1.0 and D∗ = 34.2) for the
metabolic network data.
4
Conclusions
We have shown that it is possible to use standard statistical methods in order
to determine which probability model describes the degree distribution best. We
find that the common practice of fitting a pure power-law to such experimental
network data[13, 3] may obscure information contained in the degree distribution
of biological networks. This is often done by identifying a range of connectivities
from the log-log plots of the degree distribution which can then be fitted by a
straight line. Not only is this wasteful in the sense that not all of the data is
used but it may obfuscate real, especially finite-size, trends. The same will very
likely hold true for other biological networks, too [17]. The approach used here,
on the other hand, (i) uses all the data, and (ii) can be extended to assessing
levels of confidence through combining a bootstrap procedure with the Akaike
weights. What we have shown here, in summary, is that statistical methods can
be usefully applied to protein interaction and metabolic network data.
Even though the degree distribution does not capture all (or even most) of
the characteristics of real biological networks there is reason to reevaluate previous studies. We find that formally real biological networks provide very little
support for the common notion that biological networks are scale-free. Other
fat-tailed probability distributions provide qualitatively and quantitatively better descriptions of the degree distributions of biological networks. For protein
interaction networks we found that the log-normal and the stretched exponential offer superior descriptions of the degree distribution than the powerlaw or
its finite size versions. For metabolic versions our results confirms this. Even the
exponential model outperformed the scale-free model in describing the empirical
degree distribution (we note that randomly growing networks are characterized
by an exponentially decreasing degree distribution). The best models are all
fat-tailed —like the scale-free models— but are not formally scale-free. Unfortunately, there is as yet no known physical model for network growth processes
that would give rise to log-normal or stretched exponential degree distributions.
There is thus a need to develop and study theoretical models of network
growth that are better able to describe the structure of existing networks. This
probably needs to be done in light of at least three constraints: (i) real networks are finite sized and thus, in the terms of statistical physics, mesoscopic
systems; (ii) present network data are really only samples from much larger net-
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M.P.H. Stumpf et al.
works as not all proteins are included in present experimental setup (in our case
S.cerevisiae has the highest fraction, 4773 out of approximately 5500-6000 proteins); the sampling properties have recently been studied and it was found that
generally the degree distribution of a subnet will differ from that of the whole
network. This is particularly true for scale-free networks. (iii) biological networks
are under a number of functional and evolutionary constraints and proteins are
more likely to interact with proteins in the same cellular compartment or those
involved in the same biological process. This modularity —and the information
already availabe e.g. in gene ontologies— needs to be considered. Finally there
is an additional caveat: biological networks are not static but likely to change
during development. More dynamic structures may be required to deal with this
type of problem.
Quite generally we believe that we are now at a stage where simple models do
not necessarily describe the data collected from complex processes to the extent
that we would like them to. But as Burda, Diaz-Correia and Krzywicki point
out [9], even if a mechanistic model is not correct in detail, a corresponding
statistical ensemble may nevertheless offer important insights. We believe that
the statistical models employed here will also be useful in helping to identify
more realistic ensembles.
The maximum likelihood, goodness of fit and other tools and methods for
the analysis of network data are implemented in the NetZ R-package which is
available from the corresponding author on request.
Acknowledgements. We thank the Wellcome Trust for a research fellowship
(MPHS) and a research studentship (PJI). CW is supported by the Danish Cancer Society. Financial support from the Royal Society and the Carlsberg Foundation (to MPHS and CW) is also gratefully acknowledged. We have furthermore
benefitted from discussions with Eric de Silva, Bob May and Mike Sternberg.
References
[1] I. Agrafioti, J. Swire, J. Abbott, D. Huntley, S. Butcher and M.P.H. Stumpf.
Comparative analysis of the Saccharomyces cerevisiae and Caenorhabditis elegans
protein interaction networks. BMC Evolutionary Biology, 5:23, 2005.
[2] H. Akaike. Information measures and model selection. In Proceedings of the 44th
Session of the International Statistical Institute, pages 277–291, 1983.
[3] R. Albert and A. Barabási. Statistical mechanics of complex networks. Rev. Mod.
Phys., 74:47, 2002.
[4] T.W. Anderson and D.A. Darling Asymptotic theory of certain goodness-of-fit
criteria based on stochastc processes. Ann.Math.Stat., 23:193, 1952.
[5] T.W. Anderson and D.A. Darling A test of goodness of fit J.Am.Stat.Assoc.,
49:765, 1954.
[6] A. Barabási and R. Albert. Emergence of scaling in random networks. Science,
286:509, 1999.
[7] B. Bollobás. Random Graphs. Academic Press, London, 1998.
[8] B. Bollobás and O. Riordan. Mathematical results on scale-free graphs. In S. Bornholdt and H. Schuster, editors, Handbook of Graphs and Networks, pages 1–34.
Wiley-VCH, 2003.
Statistical Model Selection Methods Applied to Biological Networks
77
[9] Z. Burda and J. Diaz-Correia and A. Krzywicki. Statistical ensemble of scale-free
random Graphs. Phys. Rev. E, 64,:046118,2001.
[10] K. Burnham and D. Anderson. Model selection and multimodel inference: A practical information-theoretic approach. Springer, 2002.
[11] D. R. Cox and Reid. A note on pseudolikelihood constructed from marginal
densities. Biometrika, 91:729, 2004.
[12] A. Davison. Statistical models. Cambridge University Press, Cambridge, 2003.
[13] S. Dorogovtsev and J. Mendes. Evolution of Networks. Oxford University Press,
Oxford, 2003.
[14] S. Dorogovtsev, J. Mendes, and A. Samukhin. Multifractal properties of growing
networks. Europhys. Lett., 57:334;cond–mat/0106142, 2002.
[15] T. Ito, K. Tashiro, S. Muta, R. Ozawa, T. Chiba, M. Nishizawa, K. Yamamoto,
S. Kuhara, and Y. Sakaki. Towards a protein-protein interaction map of the
budding yeast: A comprehensive system to examine two-hybrid interactions in all
possible combinations between the yeast proteins. PNAS, 97:1143, 2000.
[16] S. Maslov and K. Sneppen. Specificity and stability in topology or protein networks. Science, 296:910, 2002.
[17] R. May and A. Lloyd. Infection dynamics on scale-free networks. Phys. Rev. E,
64:066112, 2001.
[18] M. Newman, S. Strogatz, and D. Watts. Random graphs with arbitrary degree
distribution and their application. Phys. Rev. E, 64:026118;cond–mat/0007235,
2001.
[19] H. Qin, H. Lu, W. Wu, and W. Li. Evolution of the yeast interaction network.
PNAS, 100:12820, 2003.
[20] K. Strimmer and A. Rambaut. Inferring confidence sets of possibly misspecified
gene trees. Proc. Roy. Soc. Lond. B, 269:127, 2002.
[21] M.P.H. Stumpf, C. Wiuf and R.M. May Subnets of scale-free networks are not
scale-free: the sampling properties of random networks. PNAS, 103:4221, 2005.
[22] A. Wagner. The yeast protein interaction network evolves rapidly and contains
few redundant duplicate genes. Mol. Biol. Evol., 18:1283, 2001.
[23] I. Xenarios, D. Rice, L. Salwinski, M. Baron, E. Marcotte, and D. Eisenberg. Dip:
the database of interacting proteins. Nucl. Acid. Res., 28:289, 2000.
[24] S. Yook, Z. Oltvai, and A. Barabási. Functional and topological characterization
of protein interaction networks. Proteomics, 4:928, 2004.